Chapters 8

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Chapters 8 9

• Inference for Proportions and Two-way Tables

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When are we interested in Proportions?
When we have a categorical variable such as
gender (male/female), race (white/non-white),
quality of product (defective/non-defective).
Suppose at Auburn there are 20,000 undergraduates
and 11,000 of them are female. This means the
proportion of students that is female is given by
p X/N 11,000/20,000 0.55 We most likely
will not know the value of this population
parameter so we would estimate it with a sample
proportion, p. If in a random sample of n 100
students we count x 60 females we have p x/n
60/100 0.6. We will use this p sample
statistic to build confidence intervals and test
hypotheses about the unknown population
parameter, p.
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C.I. For Population Proportion
Suppose, before the presidential election in
2004, in a sample of 100 people, 50 said they
would vote for Bush. What is the 95 confidence
interval for the population proportion p?
p (502)/(1004)0.5
You hear these in the news all the time. 50 of
people would vote for Bush, with a sampling error
of 9.6.
1.96(.0490) .096 .5 .096 (.404 to .596)
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Macro Output for Nov. 2004 Example
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C.I. For Population ProportionGroup Work
Suppose in a sample of 1000 people, 40 said they
would vote for Kerry. What is the 95 confidence
interval for the population proportion p?
40 of people would vote for Kerry, with a
sampling error of 3. Have you noticed that
the sampling error is often 3? The sample
size is usually in the neighborhood of 1000.
p 402/1004 0.4004
1.96(.01546) .0303
.4004 .0303 (.37 to .431)
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HW 1Population Proportion of Female Students

• We have a sample that gives us a proportion of
  female students in STAT 2610. Look back at your
  output from the beginning of the semester (or do
  the EXCEL procedure again) to obtain the sample
  proportion.
• How would you estimate the true population
  proportion of female students in STAT 2610? Use
  a 95 confidence level.

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Hw 2 Population Proportion of Students who
Agreed free minutes is the most important
criterion in a cell phone plan

• We have a sample that gives a proportion of
  students in STAT 2610 who at least agreed that
  amount of free minutes is the most important cell
  phone criterion. Use EXCEL to obtain the sample
  proportion. Then compute the Std.Err of p
• Use the resulting output to estimate the true
  population proportion of students in STAT 2610
  who at least agree on the cell phone issue. Use
  a 90 confidence level.

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Hypothesis Tests For Population Proportion
To formally test the hypothesis that p is some
value po, we use Ho p po vs. Ha p ? po, (or
p gt po or p lt po). Since the hypothesis is
evaluated under the assumption that the null is
true, we have a known proportion value to compute
the standard deviation of the sample p. So we
use a z- (instead of a t-) statistic to
incorporate the known standard deviation,
sqrtpo(1-po)/n
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Is Voter Preference Independent of Gender?
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Two-Way Tables
.45
.45.52 1200 280.8
.55
.52
.48
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Two-Way Tables
.45
259.2
280.8
343.2
.55
316.8
.52
.48
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Two-Way Tables
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Two-Way Tables

• P-value of test statistic 25.17 is lt .0005 which
  is lt a ( say, .01)
• So we REJECT Ho in favor of Ha and conclude that
  Voting for Bush or Kerry is NOT independent of
  whether someone is male or female. Voting for
  Bush or Kerry depends on gender.

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Macro Generated Two-Way Table